When you make a purchase of 1100 Gran or more from the Cashier tab, along with your Gran your account will be upgraded with Premium Features at no additional cost. The Premium Features include double the number of slots you have for Card Files under the FILE menu from 10 to 20. They also include extra space in your in-game mail(double) and additional theater functionality, which allows you to watch Replays of your battles, and share them.

When you finish a battle, you will be given the option to Save it. Select a Saved Battle Slot from the dropdown menu in the Theater Tab and hit Save Battle. You may also type in a name that will help you remember it later.
If you want to share your battle, just select Display URL and cut and paste it wherever you want. The system will automatically disguise your Rival Iczer's name and Avatar for other users, although you will be able to see them when you play them directly from the menu.
You begin with 5 slots to save battle Replays. You may purchase more for 100 Gran each. Unlike the Premium Upgrade, additional slots can be bought with gran earned outside of the Cashier tab.
When you playback a battle, you will be given controls like a DVD player, where you can skip forward or back through the turns, pause or jump to the beginning or end of the battle.